## Learning objectives

[knowledge and understanding]

Know, understand and be able to explain all the essential arguments in the section "Programma esteso" below (except for the proofs of those marked (**) and (***), but see "Modalità di verifica dell'apprendimento" below), which form a strong theoretical understanding of stochastic processes.

[applying knowledge and understanding]

Be able to solve exercises and problems on the course arguments, in particular all the "homeworks" assigned during the lessons.

[making judgements]

Be able to check whether a process is well-defined and when it enjoys the properties introduced in the lectures (i.e. be adapted, a semimartingale, M2-loc class, etc.).

[learning skills]

Be able to read and understand scientific texts which build on the knowledge of continuous-time stochastic processes, stochastic integration and stochastic differential equations in dimension 1.

## Prerequisites

Measure spaces, probability spaces, Borel-Cantelli lemmas, random variables, mathematical expectation, modes of convergence for random variables, L^p spaces

## Course unit content

In the first part of the course we introduce continuous-time stochastic processes and we deal with the new issues arising from this object. In particular, we develop the tools needed for the study of stochastic processes and we show the existence of the Brownian motion.

Second part is devoted to the construction of the stochastic integral and to the study of its properties, in particular through martingales.

In the third part we give a short introduction to stochastic differential equations.

## Full programme

The list below is based on what was done in the recent years in this course and is subject to small changes and adaptations for the needs of the present year. An unpdated list will be given to the students at the end of the course.

1. Bernoulli process

2. Random walks

3. Markov chains

4. Poisson process

5. Brownian motion

6. A process is measurable with respect to the produict σ-algebra iff its components are (*)

7. Finite dimensional laws determine the law as a process (*)

8. Countable times process: modification => indistinguishable

9. Continuous trajectories process: modification => indistinguishable (*)

10. BM is a centered Gaussian process with c(s,t)=min(s,t) (*)

11. A centered Gaussiano process with c(s,t)=min(s,t) => 1) 2) 3) of BM (*)

12. Hypothesis of extension thm => compatible finite-dimensional laws (**)

13. Kolmogorov Extension Theorem (**)

14. Kolmogorov Regularity Theorem (***)

15. Existence of BM (*)

16. Invariance properties of BM (**)

17. Total variation of a function, BV functions, integrals

18. Quadratic variation of a stochastic process

19. Quadratic variation of BM (*)

20. A.s. convergence of approximants to quadratic variation for increasing partitions (**)

21. Trajectories of BM are not BV and not Hoelder for alpha>1/2 (*)

22. Filtrations, natural filtrations, right-continuous filtrations, adapted processes, progressive processes

23. BM is always a BM with respects to its own natural filtration (*)

24. Progressive => adapted

25. Adapted + right- or left-continuous trajectories => progressive (**)

26. M² processes, S² processes, stochastic integral I for S² processes

27. I is linear and it is an isometry on S² (*)

28. I extends uniquely to M² (*)

29. S² is dense in M² (***)

30. Mean and variance of I(X) (*)

31. Condizionale expectation and second conditional moment of I(X) (including the two lemmas) (**)

32. I(X) is an L²-mg, its quadratic variation

33. Continuous-time martingales (super and sub), stopping times (s.t.)

34. Two cases in which a hitting time is a s.t. (**)

35. σ-algebra of a s.t. and measurability of a progressive X when evaluated at a s.t. (*)

36. Downcrossing lemma (*)

37. Doob's backward convergece thm (**)

38. Countable times sub-mgs: 1) if extrema of T are included, then L¹-norm is bounded (*)

39. Countable times sub-mgs: 2) if right estremum is included and L¹-norm is bounded, then it is UI (**)

40. A mg admits a modification with cadlag trajectories (*)

41. Optional sampling thm (continuous) (**)

42. A stopped sub-mg is itself a sub-mg (*)

43. Maximal inequality (*)

44. I(X) has a continuous modification which is a.s. uniform limit of continuous approximants (*)

45. Localization thm for I on M² (*)

46. Stochastic integral for M²-loc processes (**)

47. I on M²-loc is a local mg

48. Continuity of I on M²loc (specific topologies) (**)

49. A continuos, BV mg is costant (**)

50. Doob's Lp inequality

51. Existence and characterization thm for the quadratic variation of continuous, bounded mgs (***)

52. Stopping at a s.t. commutes with the quadratic variation

53. Quadratic variation thm for local mgs (**)

54. Quadratic variation for the stochatic integral of M²-loc processes (*)

55. Quadratic variation of a semimartingale

56. Stochastic integral with an Ito process as the integrator

57. Ito formulas in dimension 1 and greater; multidimensional BM

58. Proof of Ito formula in the basic case (***)

59. Geometric BM

60. Ornstein-Uhlenbeck process

61. Stocastic differential equations (SDE), strong and weak solutions, uniqueness pathwise and in law

62. Well-posedness thm for SDEs in the Lipshitz case (***)

63. Poisson process, elementary construction and marginal law

64. Poisson process, as a limit of Bernoulli processes (**)

65. Implicit definition of Poisson and Lévy processes

66. Renewal thm for Lévy processes (**)

67. Poisson process has finite moments and independent times (*)

68. Characterization thm for Poisson process (***)

Remark. The proofs presented in the lessons must be studied at different levels, according to their complexity:

- the simplest and most immediate ones, that can be thought of as simple verifications, must be always known

- the other ones are marked with (*), (**) or (***) according to their complexity: the ones marked (*) must be always known; the ones marked (**) and (***) are in general not part of the exam, but for a small set of them fixed by the teacher some days before the examination (at most 3 af which at most one (***))

- definitions and statements of theorems must be always knowni; if any proof is not marked with (*), (**) or (***), it must be known

- homework given during the lessons count as (**)

## Bibliography

Francesco Morandin - Lecture notes 2016, 2018 and 2020

Francesco Morandin - Lecture notes 2021 (developed during the course and available online after each lesson)

Francesco Caravenna - Moto browniano e analisi stocastica

Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion

Ioannis Karatzas, Steven E. Shreve - Brownian Motion and Stochastic Calculus

David Williams - Probability with Martingales

Paolo Baldi - Equazioni differenziali stocastiche e applicazioni

Bernt Øksendal - Stochastic Differential Equations: An Introduction with Applications

## Teaching methods

Traditional classes (48 hours). Arguments are presented in a formal way, with proofs for most statements. Much stress is given to the motivations and we include some examples of applications. There are no exercise sessions scheduled, but homework is regularly assigned during lessons and students are encouraged to do it at home and possibly ask for solutions during the teacher office hours.

## Assessment methods and criteria

The examination is in the form of an interview, based on a program given by the teacher at least 3 days before. The program is formed by:

A) one exercise or problem

B) two arguments marked (**) or (***) from the updated "programma esteso"

C) one homework

The arguments in the updated "programma esteso" which are neither market (**) nor (***) can be asked for without notice, as well as all the definitions and the statements of the theorems.

Parts B) and C) can be substituted as a whole by choosing one of the advanced arguments from a list that will be made available by the teacher.

To pass the exam the student should master the mathematical language and formalism. She must know the mathematical objects and the theoretical results of the course and she should be able to use them with ease. She should also be able to prove theorems by herself.

## Other information

There will be an e-learning website, where the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC